2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74075We study Graver test sets for families of linear multi-stage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many ``building blocks'', independent of the number of scenarios, and we give an effective procedure to compute these building blocks. The paper includes an introduction to Nash-Williams' theory of better-quasi-orderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.36 ppOptimization and ControlCombinatorics90C15; 90C10; 06A06; 13P10text